The current i in an induction coil varies with time according to the graph shown in figure. Which of the following graph shows induced emf in the coil with time:

Options

A.
B.
C.
D.

Explanation

To solve the problem, we need to analyze the relationship between the current \( I \) in the induction coil and the induced electromotive force (emf) \( \epsilon \) over time. The induced emf is given by Faraday's law of electromagnetic induction, which states:

\[ \epsilon = -L \frac{dI}{dt} \]

where \( L \) is the inductance of the coil and \( \frac{dI}{dt} \) is the rate of change of current with respect to time.

### Step-by-Step Solution:

1. **Analyze the Current Graph:**
- The current \( I \) varies with time according to a given graph. We need to identify the intervals where the current is changing and where it is constant.

2. **Determine \( \frac{dI}{dt} \):**
- For the time interval from \( 0 \) to \( T_1 \):
- The current is constant (flat line), so \( \frac{dI}{dt} = 0 \).
- Thus, the induced emf \( \epsilon = -L \cdot 0 = 0 \).
- For the time interval from \( T_1 \) to \( T_2 \):
- The current is decreasing (the graph slopes downward).
- Here, \( \frac{dI}{dt} < 0 \) (negative value), which means \( \epsilon \) will be positive (since the negative sign in the formula will cancel out the negative rate of change).
- Since the slope is constant, \( \epsilon \) will also be constant during this interval.

3. **Evaluate the Remaining Time Intervals:**
- For the time interval from \( T_2 \) to \( T_3 \):
- If the current is again constant, \( \frac{dI}{dt} = 0 \) and thus \( \epsilon = 0 \).
- For the time interval from \( T_3 \) onwards:
- If the current is increasing, \( \frac{dI}{dt} > 0 \) which will result in a negative induced emf (since the formula has a negative sign).

4. **Construct the Induced EMF Graph:**
- From \( 0 \) to \( T_1 \): \( \epsilon = 0 \).
- From \( T_1 \) to \( T_2 \): \( \epsilon \) is a constant positive value.
- From \( T_2 \) to \( T_3 \): \( \epsilon = 0 \).
- From \( T_3 \) onwards: \( \epsilon \) is a constant negative value.

5. **Select the Correct Graph:**
- Based on the analysis, the graph of induced emf \( \epsilon \) versus time should reflect these intervals: zero, a constant positive value, back to zero, and then a constant negative value.

### Conclusion:
The correct graph showing the induced emf in the coil with time would be the one that matches the described behavior of \( \epsilon \).